Secure transmission of data is becoming increasingly important in society. Personal data, commercial & financial information and cryptographic keys themselves are transmitted between different locations, and it is desirable for there to be minimal (or preferably no) risk of interception. Various encryption schemes have been proposed to protect transmitted data.
An example of such a scheme is quantum cryptography, which in principle can provide completely secure transmission. Whereas most recent encryption methods rely on the difficulty of computing certain mathematical functions, quantum cryptography is based on physical phenomena. The usual goal of quantum cryptography is to share a random data string, for use as a key in the encryption (and decryption) of sensitive messages; the encryption itself is usually carried out using a suitable algorithm. The encrypted message may then safely be transmitted over an open (non-secure) communications channel.
Quantum theory tells us that measurement of an observable in a system will disturb the system, in particular where two observables are described by non-commuting operators. An example of two such observables are the polarisation states of a photon, for example, on the one hand, the vertical/horizontal states, and, on the other hand, the 45 degree/135 degree diagonal states. Quantum systems can be entangled, whereby the quantum states of two or more objects are linked, and remain linked even when the objects are separated from each other, even by considerable distances.
Those phenomena enable the construction of quantum communication systems that detect any attempt at eavesdropping, and, by rejecting any data contaminated by eavesdropping, allow the transmission of a key under demonstrably secure conditions. Thus, in one class of quantum encryption systems, photon polarisation is used to construct the key, and bits contaminated by eavesdropping are rejected.
In another class, entangled photons are used. One of an entangled pair of photons is transmitted to a receiver. Measurements are performed on the entangled photons, at the transmitter and receiver ends of the system, and the results of those experiments are used to construct a secure key.
However, at present, quantum cryptography suffers from many limitations, such as distance (due to a need for transmission of single photons), low bit rate, susceptibility to jamming, and the considerable difficulty of implementing practical, working systems.
Another field of research that utilises quantum mechanical effects is the field of quantum computing. In quantum computing, the phenomena of quantum physics, such as entanglement, are used to perform operations on data. A drawback of using the polarisation of light for quantum computation is that the polarisation of light is only suitable for single-bit computation, i.e. any polarisation state of light can be described as a superposition of two orthogonal polarisation states. E. J. Galvez et al. describe (in “Geometric phase associated with mode transformation of optical beams bearing orbital angular momentum”, Physical Review Letters, 90(20), 203901, 2003) a device for use in N-bit quantum computation.
To understand the operation of the Galvez device, one should first consider a Poincaré Sphere in Stokes space (FIG. 1). Conventionally, such a Poincaré Sphere is used to represent the state of polarisation of a beam of light.
In 3D Stokes space, polarisation is described by three mutually orthogonal co-ordinates, S1, S2 and S3, which form the three Stokes vectors. Any polarisation can be expressed as a combination of the three Stokes vectors. For a completely polarised signal or light beam, the intensity of the light S0 is given by the expression S02=S12+S22+S32, which for constant intensity is the equation of a sphere; hence, the possible polarisations of a constant intensity light beam form the locus of the surface of a sphere in Stokes space.
Polarisation is equivalent to quantum mechanical spin, and so has dimensions of angular momentum; hence M. J. Padgett and J. Courtial teach (in “Poincaré-sphere equivalent for light beams containing orbital angular momentum”, Optics Letters, 24(7), p430, 1999) that the Poincaré sphere can also be used to represent the states of orbital angular momentum of a light beam, which form a set of modes analogous to that of polarisation.
The higher-order Hermite-Gaussian (HG) and Laguerre-Gaussian (LG) modes are represented on Poincaré sphere 20 of FIG. 2. The HG modes are analogous to linear polarisations and lie on the equator of the Poincaré Sphere, whilst the two LG modes (analogous to circular polarisations) lie at the poles of the Poincaré Sphere.
For modes or groups obeying the symmetries of the Poincaré Sphere, closed contour paths (that is, a set of transformation operations taking the mode or group away from and back to an initial state) on the surface of the Poincaré Sphere result in a geometrical phase, given by the solid angle subtended by the closed contour path at the centre of the sphere. That geometric phase effect is a phase accumulation due to topological effects. Modes or groups which do not exhibit the appropriate symmetries do not acquire a geometric phase when undergoing the same set of transformation operations.
The solid angle subtended by a closed contour is given by a double integral over the azimuthal angle 2θ and the elevation angle φ (FIG. 2) [NB: the azimuthal angle θ in real space is transformed to 2θ in Stokes space.
A closed contour path A-B-C is shown on the Poincaré Sphere 20 of FIG. 2. The starting position is the south pole, the mode LG0+1. In a first transformation, the elevation angle φ is increased by π/2 so that the mode traverses the path A to lie on the equator and (for convenience of explanation—the precise HG mode is arbitrary) becomes the mode HG01. Next, in a second transformation, the azimuthal angle in real space is rotated by θ (corresponding to 2θ on the Poincaré Sphere) so that the mode, whilst still lying on the equator, traverses the locus B and becomes in general a mixed state of HG01 and HG10 modes, i.e. HG01 cos θ+HG10 sin θ. Finally, the elevation angle φ is reduced, by −π/2, so that the mode follows the path C, reaching the south pole again, and has completed a closed contour locus whilst acquiring an associated geometric phase Ω.
FIG. 3 (which is essentially the Galvez quantum-computing device) indicates how such azimuthal and elevation angles changes may be achieved in practice. The device can be considered to be made up of three portions, a portion 30 for generating the orbital-angular-momentum modes, a portion 40 for adding the geometric phase, and a portion 50 for detecting the added geometric phase.
The portion 40 for adding geometric phase is made up from (FIG. 4) a first pair of cylindrical lenses 60 with parallel axes, first and second Dove prisms 70, 80, and a second pair of parallel cylindrical lenses. All of the components are mounted on a common optical axis 100, so that light passes through each of them in sequence. The cylindrical lenses 60, 90 are mounted on actuated rotation stages, so that they may each be rotated through an angle about the optical axis 100. The first Dove prism 70 is mounted on an actuated rotation stage, so that it may be rotated through an angle θ/2 about an axis perpendicular to the optic axis 100.
Axial rotation of the first pair of cylindrical lenses 60 by an angle π/4 causes the required π/2 phase change in the elevation angle φ, and converts between LG and HG modes. Axial rotation of the second pair of cylindrical lenses 90 converts the modes back again. The first Dove prism is axially orientated at an angle θ/2 with respect to the second Dove prism, so as to spatially rotate the mode by an angle θ [NB: as previously discussed, this causes an angle 2θ in the associated Stokes Space].
FIG. 5 shows how the multiplexing of the two modes of light is achieved: the first mode which experiences the geometric phase modulations, and the second mode which does not undergo any geometric phase changes. The portion 30 for generating the orbital-angular-momentum modes is made up from a computer generated hologram (CGH) exhibiting a single phase line dislocation (FIG. 5(b)—essentially a phase singularity at its centre), a pair of beamsplitters BS1, BS2, a pair of mirrors M1 and M2, and a pair of iris stops.
The CGH is placed before a collimated lowest-order Gaussian beam of light, i.e. a LG00 mode. The CGH acts to diffract the light into multiple orders. The 0th-order (undiffracted) light is the same as the incident beam and remains a lowest order LG00 mode of light. The first-order diffracted light modes are the LG0+1 and LG0−1 modes, constituting the +1st and −1st orders respectively. The diffracted orders are passed into a power beam splitting cube BS1, and the desired orders in each of the consequent two optical paths are selected by spatial filtering, i.e. by means of an iris stop in each path. The 0th-order light is allowed through in the first path (reflecting from mirror M1); whilst only the +1st-order diffracted light is allowed through in the second path (reflecting from mirror M2). The two spatially filtered beams are reflected respectively by mirrors M1 and M2 onto a second power beam splitting cube BS2, which acts to combine the two selected modes, LG00 and LG0+1 together. The two modes then pass to the portion 40 for adding geometric phase (FIG. 4). The LG00 mode experiences no geometrical phase modulation here, whereas the higher OAM mode LG0+1 is geometrically phase modulated as described above.
As described in the Galvez paper, the geometric phase is detected (in the portion 50 for detecting the added geometric phase) by directly imaging onto a CCD camera the output from the portion 40 for adding geometric phase, and noting the relative rotation of the interference pattern resulting from the superposition of the LG00 and LG0+1 modes.
The present invention seeks to provide an encryption method and apparatus in which at least some of the above-mentioned problems with prior-art encryption systems are ameliorated.